WITH A CENSUS OF THE AMPHICHEIRALS WITH TWELVE CROSSINGS. 239 
schemes is made clear by a consideration of the representation in figs. 3 and 3' of a 
reversible tangle R», where the crossings 7 1? y 2 , ■ ■ ■ . y n +i are supposed to lie within 
the circle drawn in both figures. By the application of the distortion D*, the alpha- 
betical scheme becomes in the first case 
or 
* 7i 72 • • • • 7«+i y .... p a y t .... 7j a q 
-+- 
x _ 7i 72 • • • • 7*+i y • • • • ? a yj . . . . a p 
and in the second case 
or 
* Yi 72 ■ • ■ • Y«+i a p .... y yi .... 7j a q 
x 7 i 72 • • • • y«+i a p .... q a 7j ... 7i y 
Within a reversible tangle R„, may exist the possibility of distortions D„_j, D„_ 2 , 
.... D 2 , Dx, Do, which may be applied singly or in combination with others. A knot 
is invariant under a distortion D 0 , since by it, the two threads of the reversible tangle 
R 0 are untwisted at a point a preceding y x , to be twisted at a point a' beyond 71 , and 
consequently the general arrangement of the crossings is undisturbed. Hence in a 
consideration of the different forms of a given scheme the distortion of lowest 
order to be considered is the distortion Dj. The reason that makes the consideration 
of D 0 unnecessary applies also to a similar distortion Dl on the sequence 
• • • • 7l 72 • • • • y»+2 • • • ■ 7l 72 • • ■ • 7n+2 ■ ■ ■ 
or 
• • • ■ 7i 72 • • • • 7"+2 • • • • y»+2 7»+! 7i • • ■ • 
Also it is unnecessary to consider a distortion affecting more than one-half of 
the total number of crossings, since it is equivalent to a distortion applied to 
the remainder of the knot. Hence the different knots of orders 3, 4, and 5 have 
but one form. 
